Your Sony T1 is one of those camers made with bad CCDs. That is the image sensor has known defect after production and market launch. Instead of recall their products, Sony offer owners to send/bring in their camera for CCD replacement without service/parts charge. Go to Sony web site to find a Sony authorized service centre nearest to wher you live, phone them up to confirm free service. You may either mail in or bring it to them in person. As far as I know, Sony will reimburse the shipping cost to you as well as sending the fixed camera for free.
Take a look of a third party comment/information regarding the bad CCDs by clicking here before taking further action. It didn't show black screen as a symptom, but I Got mine fixed for free with the black screen, I rather say very, very dark screen.

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Hi,It is known problem of ccd imager chip failure in DSC-T1 camera, ccd imager chip will be replace to get correct live view in camera mode to take fresh pictures.Local camera repair shop will do this job because camera will take apart, let me know to provide you part number info if you are handy with electronics stuff repair and willing to replace ccd imager chip.Hope this will help.Thanks.

Hi,This is known problem of ccd imager chip failure in DSC-T1 camera and ccd imager chip will be replaced to get live view in LCD to take pictures.Sony offers free of cost repair for DSC-T1 for ccd imager failure problem.Click Sony CCD Imager Support for details.Hope this info will help you to get it fixed.Thanks.

Hi. This is a common fault on the DSC-T1 and is due to a faulty CCD. If you take it to a sony shop, they will send it to sony for a free repair (mention to them it is a common fault, and that you expect a free repair). They will send it off for you, and in a few days you will get your workingcamera back.

It is known problem of bad ccd imager in Sony DSC-T1 Cyebershot camera and ccd imager will be replaced.Sony will repair this camera free of cost for ccd imager problem as per announced scheme. You will be charged $20 for shipment.Click Sony CCD Imager Support fro details.